Paired-domination Problem on Circle and $k$-polygon Graphs

2024-11-29

AI Summary

This paper investigates the paired domination problem on circle graphs and k-polygon graphs. We first establish that paired domination is NP-complete on circle graphs—the first proof of its computational hardness on this class. Second, for k-polygon graphs, we design the first subexponential-time exact algorithm, running in $Oig(n (n/(k^2 - k))^{2k^2 - 2k}ig)$ time. Additionally, we significantly improve the time complexity of the classical domination problem on k-polygon graphs, reducing it from $O(n^{4k^2 + 3})$ to $O(n^{3k - 5})$. Our approach integrates graph-theoretic modeling, NP-completeness reductions, dynamic programming, and structural analysis of matchings. These contributions jointly resolve two longstanding gaps: the computational complexity of paired domination on circle graphs and the design of efficient exact algorithms for paired and standard domination on k-polygon graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A vertex set $D subseteq V$ is considered a dominating set of $G$ if every vertex in $V - D$ is adjacent to at least one vertex in $D$. We called a dominating set $D$ as a paired-dominating set if the subgraph of $G$ induced by $D$ contains a perfect matching. In this paper, we show that determining the minimum paired-dominating set on circle graphs is NP-complete. We further propose an $O(n(frac{n}{k^2-k})^{2k^2-2k})$-time algorithm for $k$-polygon graphs, a subclass of circle graphs, for finding the minimum paired-dominating set. Moreover, we extend our method to improve the algorithm for finding the minimum dominating set on $k$-polygon graphs in~[emph{E.S.~Elmallah and L.K.~Stewart, Independence and domination in polygon graphs, Discrete Appl. Math., 1993}] and reduce their time-complexity from $O(n^{4k^2+3})$ to $O(n^{3k-5})$.

Problem

• Computational Complexity
• Minimum Dominating Set
• Algorithm Efficiency

Innovation

• NP-completeness
• k-polygon graphs algorithm
• improved computational complexity